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04-10-2016 06:09 PM

Running a quasi-likelihood beta regression but having difficulty with the intercept term that comes out in my regressions...

Dependent variable can only take on values between 0.10-1; dependent variable is also left-skewed and there is a spike at 1. So using quasi-likelihood regressions with a beta distribution.

The intercept term comes out above 1.30 when I run my models - which doesn't make sense. Wondering if there is something wrong with my code:

PROC GLIMMIX DATA = TEST;

MODEL QOL = income age ...

/LINK = LOGIT S DIST = BETA;

output out=fracout pred(ilink)=pred lcl(ilink)=lower ucl(ilink)=upper;

RANDOM _RESIDUAL_;

WHERE QOL GT 0;

weight wtfa_sa;

RUN;

Any suggestions greatly appreciated.

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04-11-2016 08:40 AM

Well, the intercept would be the logit of the value if all of your IV values were zero. Putting the value back on the original scale, I get intercept (orig scale) = exp(1.3)/(1+exp(1.3)) = 0.7858. Is that close to what you would expect?

Steve Denham

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04-11-2016 11:56 AM

Thanks! That is closer to what I am expecting. But would that change the interpretation of the coefficients of the variables that I seek to analyze?

Here is some of my output (in this model the intercept was 1.20, others it's 1.3 or ranges between):

INTERCEPT 1.2091 | 0.005849 | 137E3 | 206.73 | <.0001 |

LESS_HIGH 0.003781 | 0.006926 | 137E3 | 0.55 | 0.5852 |

SOME_COLLEGE 0.02738 | 0.007094 | 137E3 | 3.86 | 0.0001 |

INCOME_CAT 0.003528 | 0.006746 | 137E3 | 0.52 | 0.6010 |

FEMALE 0.05704 | 0.005573 | 137E3 | 10.24 | <.0001 |

26042 | 93.2452 | . | . | . |

1.3039 | . | . | . | . |

Say looking at the 'female' dummy variable (coeffiicent is next to the variable name)...would the interpretation be like that of an OLS model - being female leads to a 0.06 higher quality of life? Or would there be a need to transform as you have done with the intercept term?

I do believe dist = beta is the right choice, but am concerned with the interpretation of the coefficients.

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04-11-2016 02:16 PM

The coefficient implies that a one unit change in the variable leads to a change in the logit equal to the coefficient, with all other values at their mean. To calculate a change on the original scale, you would need to plug the coefficients times the mean values for all other variables, and then calculate the logit for female=0 and female=1, using that coefficient. Then apply the inverse transformation for the logit to get the value on the original scale. The coefficients can't be interpreted exactly as those in an OLS regression because of the logistic link function implied by the selection of the beta distribution.

Steve Denham